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I am running simulations, and the output will be a line y = mx+b. I am interested in the area below the line between x=0 and x=1. I am only interested in the area that is below the diagonal y = x.

I have figured out how to determine this area by finding areas of triangles. But to do so, I have to define 6 cases. This requires many if-else statements in my computer program, and is inefficient.

I was wondering if there is an elegant solution to this problem which will not require such a complex program?

In the diagram below, the diagonal line is solid, my line of interest is the dotted line.

enter image description here

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Let $y = mx+b$ be the equation of your line, and then find the point of intersection with $y=x$, which will be $(p,p)$.

Then you need to find two integrals: the integral from $0$ to $p$ of $x-(mx+b)$ and the integral from $p$ to $1$ of the same function. Then pick whichever one is nonnegative!

EDIT: Just realized the lines don't have intersect. If that happens, just take the integral from $0$ to $1$ of $x-(mx+b)$ and if it's negative, return 0.

I guess I only simplified a few cases.

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